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Let’s check out your problem:
g
(
x
)
=
∫
2
x
13
1
+
t
2
d
t
g
′
(
5
)
=
\begin{array}{l}g(x)=\int_{2}^{x} \frac{13}{1+t^{2}} d t \\ g^{\prime}(5)=\end{array}
g
(
x
)
=
∫
2
x
1
+
t
2
13
d
t
g
′
(
5
)
=
View step-by-step help
Home
Math Problems
Calculus
Find indefinite integrals using the substitution
Full solution
Q.
g
(
x
)
=
∫
2
x
13
1
+
t
2
d
t
g
′
(
5
)
=
\begin{array}{l}g(x)=\int_{2}^{x} \frac{13}{1+t^{2}} d t \\ g^{\prime}(5)=\end{array}
g
(
x
)
=
∫
2
x
1
+
t
2
13
d
t
g
′
(
5
)
=
Apply Fundamental Theorem of Calculus:
We use the Fundamental Theorem of Calculus which says that if
g
(
x
)
=
∫
a
x
f
(
t
)
d
t
g(x) = \int_{a}^{x}f(t)\,dt
g
(
x
)
=
∫
a
x
f
(
t
)
d
t
, then
g
′
(
x
)
=
f
(
x
)
g'(x) = f(x)
g
′
(
x
)
=
f
(
x
)
.
Calculate
g
′
(
x
)
g'(x)
g
′
(
x
)
:
So,
g
′
(
x
)
=
13
1
+
x
2
g'(x) = \frac{13}{1+x^{2}}
g
′
(
x
)
=
1
+
x
2
13
.
Find
g
′
(
5
)
g'(5)
g
′
(
5
)
:
Now we need to find
g
′
(
5
)
g'(5)
g
′
(
5
)
, which means we plug in
x
=
5
x=5
x
=
5
into
g
′
(
x
)
g'(x)
g
′
(
x
)
.
Substitute
x
=
5
x=5
x
=
5
:
g
′
(
5
)
=
13
1
+
5
2
g'(5) = \frac{13}{1+5^{2}}
g
′
(
5
)
=
1
+
5
2
13
.
Simplify Result:
Calculate
g
′
(
5
)
=
13
1
+
25
g'(5) = \frac{13}{1+25}
g
′
(
5
)
=
1
+
25
13
.
Simplify Result:
Calculate
g
′
(
5
)
=
13
1
+
25
g'(5) = \frac{13}{1+25}
g
′
(
5
)
=
1
+
25
13
.
g
′
(
5
)
=
13
26
g'(5) = \frac{13}{26}
g
′
(
5
)
=
26
13
.
Simplify Result:
Calculate
g
′
(
5
)
=
13
1
+
25
g'(5) = \frac{13}{1+25}
g
′
(
5
)
=
1
+
25
13
.
g
′
(
5
)
=
13
26
g'(5) = \frac{13}{26}
g
′
(
5
)
=
26
13
. Simplify
g
′
(
5
)
=
1
2
g'(5) = \frac{1}{2}
g
′
(
5
)
=
2
1
.
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Question
Find
d
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d
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[
5
ln
(
x
5
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]
\frac{d^{2}}{d x^{2}}\left[5 \ln \left(x^{5}\right)\right]
d
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Consider the curve given by the equation
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Consider the curve given by the equation
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(\square
(
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Question
Can this differential equation be solved using separation of variables?
\newline
d
y
d
x
=
3
x
2
y
−
8
y
\frac{d y}{d x}=\frac{3}{x^{2} y-8 y}
d
x
d
y
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2
y
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3
\newline
Choose
1
1
1
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\newline
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Posted 1 year ago
Question
y
=
x
4
d
y
d
x
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y
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d
x
d
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Question
Solve
∫
d
x
4
sin
2
(
x
)
+
8
sin
(
2
x
)
−
3
cos
2
(
x
)
\int \frac{dx}{4\sin^2(x)+8\sin(2x)-3\cos^2(x)}
∫
4
s
i
n
2
(
x
)
+
8
s
i
n
(
2
x
)
−
3
c
o
s
2
(
x
)
d
x
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Posted 1 year ago
Question
(e) Use the substitution
\newline
x
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θ
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x
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3
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\newline
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3
2
d
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−
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2
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3
2
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∫
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−
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x
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Posted 1 year ago
Question
Evaluate the integral
∫
x
+
3
4
x
+
4
d
x
\int \frac{x+3}{4 x+4} d x
∫
4
x
+
4
x
+
3
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
4
x
+
ln
∣
x
+
1
∣
+
C
\frac{1}{4} x+\ln |x+1|+C
4
1
x
+
ln
∣
x
+
1∣
+
C
\newline
(B)
1
4
x
+
2
ln
∣
x
+
1
∣
+
C
\frac{1}{4} x+2 \ln |x+1|+C
4
1
x
+
2
ln
∣
x
+
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+
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\newline
(C)
1
4
x
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ln
∣
x
+
1
∣
2
+
C
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4
1
x
+
2
l
n
∣
x
+
1∣
+
C
\newline
(D)
1
4
x
+
ln
∣
x
+
1
∣
4
+
C
\frac{1}{4} x+\frac{\ln |x+1|}{4}+C
4
1
x
+
4
l
n
∣
x
+
1∣
+
C
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Question
You pick a card at random, put it back, and then pick another card at random.
\newline
4
4
4
\newline
5
5
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\newline
6
6
6
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7
7
7
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5
5
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4
4
4
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\newline
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Question
Jeanne's bank account earns interest annually. The equation shows her starting balance of
$
500
\$500
$500
and her balance at the end of three years,
$
546.36
\$546.36
$546.36
. At what rate
r
r
r
did Jeanne earn interest?
\newline
546.36
=
500
(
1
+
r
)
546.36 = 500(1 + r)
546.36
=
500
(
1
+
r
)
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Posted 7 months ago
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